Elastic Solids to Metrology. 549 



where W and W are the weights of the flask and of the con- 

 tained liquid. We have neglected the influence of atmo- 

 spheric pressure, which is the same as in (8) or (9). For 

 comparative results we may also neglect the influence of 

 the flask's own weight, supposing it supported in an invariable 

 wav and confine our attention to the effect of the weight of 

 the liquid, as practically the only variable. For this we have 



8ii=W'(3H-4J')/3* (38) 



For given values of W, H, ?', and k, the change in the total 

 volume of the flask's material is the same for all values of that 

 volume. A thin-walled flask suffers exactly the same change 

 of volume as a thick-walled flask. This seems rather paradox- 

 ical at first sight. It simply means, however, that the mean 

 expansion per unit of volume varies inversely as the whole 

 volume. The thin-walled flask, owing to its smaller volume, 

 has the larger expansion per unit of volume, and so the 

 larger internal strains and stresses. 



The change 8v refers to the whole flask ; but in such a case 

 as that of a litre or 500 c.c. flask filled up to the mark, 8v 

 would practically apply to the portion below the mark. 



If the flask, instead of being supported, floated immersed 

 to a depth H' in the liquid, we should obtain in place of (37"), 

 neglecting atmospheric pressure as before, 



8 v =-\XZ/M-\X f/ (?>K'-41;")!n, . . (39) 

 where W" is the weight of the liquid displaced, J" the height 

 of its C.G. above the base of the flask. 



Ordinarily, internal liquid pressure will tend to increase 

 and external liquid pressure to diminish the volume of the 

 walls of the flask, If, however, ?' exceed 3H/4, and f" 

 exceed 3H'/4, the reverse will be the case. 



If the inner surface of the flask be cylindrical or cup- 

 shaped, ?' will be less than 3H/4. The two would become 

 equal if the inner surface were a cone vertex downwards, 

 and f would exceed 3H/4 for certain cusped forms of 

 surfaces. 



Thus, theoretically, a flask whose inner surface was a cone, 

 of any solid angle, vertex downwards, would have the curious 

 property that the volume of the material would be unaffected 

 by the pressure of any liquid it might contain. 



[fallow Cylinder containing or surrounded by Liquid. 



§ 12. As already stated, we cannot in general determine 

 the effect on the internal capacity of a flask due to the 

 pressure of contained liquid. We can, however, obtain this 



Phil. Mag. S. 6. Vol. 2. No. 11. Nov. 1901. 2 



